Wednesday, August 1, 2007

Kick my brains around the floor

Rebecca and I have moved our meeting time to Thursday mornings, with the result that my Wednesdays are now entirely free. Despite this [or perhaps as a result of it] I wasn't as productive today as I would have liked. I've only got two questions remaining in the assignment, but there's also question 5 still to fix. On the other hand I do have Friday morning free for writing up this time, with no more QUAN assignments due for at least two weeks.

One of the questions I did today asked us to find the conditions necessary for a certain property of derivatives to hold in the R2 -> R2 case, then added that students who have done MATH 207 and MATH 312 should extend the result to higher dimensions. Having just spent several hours banging my brains on three special cases of the same property, I did the extension with no strenuous thought. My answer was based entirely on my observations of certain patterns in the earlier questions followed by some simple extrapolation. What I can't decide is whether this is sufficient or whether a more formal derivation is expected.

More seriously, I'm not certain that my answer is actually correct. As it holds for both R2 -> R2andR3 -> R3, it seems entirely reasonable that it should hold for all higher dimensions, and I think I could probably prove this; but the proof doesn't seem to need any material from outside the course, so why wasn't this part of the question itself?

So far doing assignment 3 has been like chipping 1/4 carat diamonds out of a 1 kg block of chalk using a pin. The gems are in there, but there's a hell of a lot of pointless work required before you get anywhere near them. Questions 9 and 10 promise nothing but more of the same.